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G = C48⋊C2order 96 = 25·3

2nd semidirect product of C48 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C482C2, C162S3, C6.2D8, C31SD32, C2.4D24, C8.14D6, C4.2D12, D24.1C2, C12.25D4, Dic121C2, C24.15C22, SmallGroup(96,7)

Series: Derived Chief Lower central Upper central

C1C24 — C48⋊C2
C1C3C6C12C24D24 — C48⋊C2
C3C6C12C24 — C48⋊C2
C1C2C4C8C16

Generators and relations for C48⋊C2
 G = < a,b | a48=b2=1, bab=a23 >

24C2
12C4
12C22
8S3
6Q8
6D4
4D6
4Dic3
3Q16
3D8
2Dic6
2D12
3SD32

Character table of C48⋊C2

 class 12A2B34A4B68A8B12A12B16A16B16C16D24A24B24C24D48A48B48C48D48E48F48G48H
 size 11242224222222222222222222222
ρ1111111111111111111111111111    trivial
ρ211-111-1111111111111111111111    linear of order 2
ρ311111-111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ411-111111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ52202202-2-2220000-2-2-2-200000000    orthogonal lifted from D4
ρ6220-120-122-1-1-2-2-2-2-1-1-1-111111111    orthogonal lifted from D6
ρ7220-120-122-1-12222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ82202-20200-2-2-222-200002-2-22-2-222    orthogonal lifted from D8
ρ92202-20200-2-22-2-220000-222-222-2-2    orthogonal lifted from D8
ρ10220-120-1-2-2-1-100001111-33-333-33-3    orthogonal lifted from D12
ρ11220-120-1-2-2-1-1000011113-33-3-33-33    orthogonal lifted from D12
ρ12220-1-20-100112-2-223-3-33ζ83ζ38ζ38ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ38785ζ3ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ38ζ38    orthogonal lifted from D24
ρ13220-1-20-100112-2-22-333-3ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ38ζ38ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ38ζ38ζ87ζ38785ζ3    orthogonal lifted from D24
ρ14220-1-20-10011-222-2-333-3ζ87ζ3285ζ3285ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ32838ζ32ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285    orthogonal lifted from D24
ρ15220-1-20-10011-222-23-3-33ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ3285ζ3285ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ3285ζ3285ζ83ζ32838ζ32    orthogonal lifted from D24
ρ162-20200-2-2200ζ16131611ζ1615169ζ16716ζ16516322-2-2ζ1615169ζ165163ζ165163ζ1615169ζ16131611ζ16131611ζ16716ζ16716    complex lifted from SD32
ρ172-20200-22-200ζ16716ζ16131611ζ165163ζ1615169-2-222ζ16131611ζ1615169ζ1615169ζ16131611ζ16716ζ16716ζ165163ζ165163    complex lifted from SD32
ρ182-20200-2-2200ζ165163ζ16716ζ1615169ζ1613161122-2-2ζ16716ζ16131611ζ16131611ζ16716ζ165163ζ165163ζ1615169ζ1615169    complex lifted from SD32
ρ192-20200-22-200ζ1615169ζ165163ζ16131611ζ16716-2-222ζ165163ζ16716ζ16716ζ165163ζ1615169ζ1615169ζ16131611ζ16131611    complex lifted from SD32
ρ202-20-10012-23-3ζ16716ζ16131611ζ165163ζ1615169ζ1614ζ316141610ζ3ζ166ζ3162ζ3162ζ166ζ32166162ζ32ζ1614ζ321610ζ321610165ζ3163ζ3163ζ167ζ3216716ζ32167ζ3216ζ3216ζ165ζ3165163ζ3ζ1615ζ321615169ζ321615ζ32169ζ32169ζ1613ζ316131611ζ31613ζ31611ζ31611    complex faithful
ρ212-20-1001-22-33ζ165163ζ16716ζ1615169ζ16131611ζ166ζ32166162ζ32ζ1614ζ321610ζ321610ζ1614ζ316141610ζ3ζ166ζ3162ζ3162ζ1615ζ321615169ζ32165ζ3163ζ3163ζ165ζ3165163ζ31615ζ32169ζ321691613ζ31611ζ31611ζ1613ζ316131611ζ3167ζ3216ζ3216ζ167ζ3216716ζ32    complex faithful
ρ222-20-1001-223-3ζ16131611ζ1615169ζ16716ζ165163ζ1614ζ321610ζ321610ζ166ζ32166162ζ32ζ166ζ3162ζ3162ζ1614ζ316141610ζ3167ζ3216ζ3216ζ1613ζ316131611ζ31613ζ31611ζ31611ζ167ζ3216716ζ32ζ165ζ3165163ζ3165ζ3163ζ3163ζ1615ζ321615169ζ321615ζ32169ζ32169    complex faithful
ρ232-20-10012-2-33ζ16716ζ16131611ζ165163ζ1615169ζ166ζ3162ζ3162ζ1614ζ316141610ζ3ζ1614ζ321610ζ321610ζ166ζ32166162ζ32ζ165ζ3165163ζ3167ζ3216ζ3216ζ167ζ3216716ζ32165ζ3163ζ31631615ζ32169ζ32169ζ1615ζ321615169ζ321613ζ31611ζ31611ζ1613ζ316131611ζ3    complex faithful
ρ242-20-1001-223-3ζ165163ζ16716ζ1615169ζ16131611ζ1614ζ321610ζ321610ζ166ζ32166162ζ32ζ166ζ3162ζ3162ζ1614ζ316141610ζ31615ζ32169ζ32169ζ165ζ3165163ζ3165ζ3163ζ3163ζ1615ζ321615169ζ32ζ1613ζ316131611ζ31613ζ31611ζ31611ζ167ζ3216716ζ32167ζ3216ζ3216    complex faithful
ρ252-20-1001-22-33ζ16131611ζ1615169ζ16716ζ165163ζ166ζ32166162ζ32ζ1614ζ321610ζ321610ζ1614ζ316141610ζ3ζ166ζ3162ζ3162ζ167ζ3216716ζ321613ζ31611ζ31611ζ1613ζ316131611ζ3167ζ3216ζ3216165ζ3163ζ3163ζ165ζ3165163ζ31615ζ32169ζ32169ζ1615ζ321615169ζ32    complex faithful
ρ262-20-10012-2-33ζ1615169ζ165163ζ16131611ζ16716ζ166ζ3162ζ3162ζ1614ζ316141610ζ3ζ1614ζ321610ζ321610ζ166ζ32166162ζ32ζ1613ζ316131611ζ31615ζ32169ζ32169ζ1615ζ321615169ζ321613ζ31611ζ31611167ζ3216ζ3216ζ167ζ3216716ζ32165ζ3163ζ3163ζ165ζ3165163ζ3    complex faithful
ρ272-20-10012-23-3ζ1615169ζ165163ζ16131611ζ16716ζ1614ζ316141610ζ3ζ166ζ3162ζ3162ζ166ζ32166162ζ32ζ1614ζ321610ζ3216101613ζ31611ζ31611ζ1615ζ321615169ζ321615ζ32169ζ32169ζ1613ζ316131611ζ3ζ167ζ3216716ζ32167ζ3216ζ3216ζ165ζ3165163ζ3165ζ3163ζ3163    complex faithful

Smallest permutation representation of C48⋊C2
On 48 points
Generators in S48
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(2 24)(3 47)(4 22)(5 45)(6 20)(7 43)(8 18)(9 41)(10 16)(11 39)(12 14)(13 37)(15 35)(17 33)(19 31)(21 29)(23 27)(26 48)(28 46)(30 44)(32 42)(34 40)(36 38)

G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (2,24)(3,47)(4,22)(5,45)(6,20)(7,43)(8,18)(9,41)(10,16)(11,39)(12,14)(13,37)(15,35)(17,33)(19,31)(21,29)(23,27)(26,48)(28,46)(30,44)(32,42)(34,40)(36,38)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (2,24)(3,47)(4,22)(5,45)(6,20)(7,43)(8,18)(9,41)(10,16)(11,39)(12,14)(13,37)(15,35)(17,33)(19,31)(21,29)(23,27)(26,48)(28,46)(30,44)(32,42)(34,40)(36,38) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(2,24),(3,47),(4,22),(5,45),(6,20),(7,43),(8,18),(9,41),(10,16),(11,39),(12,14),(13,37),(15,35),(17,33),(19,31),(21,29),(23,27),(26,48),(28,46),(30,44),(32,42),(34,40),(36,38)])

C48⋊C2 is a maximal subgroup of
D487C2  C16⋊D6  C16.D6  D8⋊D6  S3×SD32  D6.2D8  Q32⋊S3  C144⋊C2  C323SD32  C24.49D6  C6.D24  D24.D5  Dic12⋊D5  C48⋊D5
C48⋊C2 is a maximal quotient of
C2.Dic24  C486C4  C2.D48  C144⋊C2  C323SD32  C24.49D6  C6.D24  D24.D5  Dic12⋊D5  C48⋊D5

Matrix representation of C48⋊C2 in GL2(𝔽23) generated by

01
122
,
129
211
G:=sub<GL(2,GF(23))| [0,1,1,22],[12,2,9,11] >;

C48⋊C2 in GAP, Magma, Sage, TeX

C_{48}\rtimes C_2
% in TeX

G:=Group("C48:C2");
// GroupNames label

G:=SmallGroup(96,7);
// by ID

G=gap.SmallGroup(96,7);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,73,79,506,50,579,69,2309]);
// Polycyclic

G:=Group<a,b|a^48=b^2=1,b*a*b=a^23>;
// generators/relations

Export

Subgroup lattice of C48⋊C2 in TeX
Character table of C48⋊C2 in TeX

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